1. Strange Attractors From Art to Science J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to the University of Wisconsin - Madison Physics Colloquium On November 14, 1997

4. Typical Experimental Data Time 0 500 x 5 -5

7. Solutions Are Seldom Chaotic Chaotic Data (Lorenz equations) Solution of model equations Chaotic Data (Lorenz equations) Solution of model equations Time 0 200 x 20 -20

8. How common is chaos? Logistic Map x n +1 = Ax n (1 - x n ) -2 4 A Lyapunov Exponent 1 -1

9. A 2-D Example (Hénon Map) 2 b -2 a -4 1 x n +1 = 1 + ax n 2 + bx n -1

10. The Hénon Attractor x n +1 = 1 - 1.4 x n 2 + 0.3 x n -1

11. Mandelbrot Set a b x n +1 = x n 2 - y n 2 + a y n +1 = 2 x n y n + b z n +1 = z n 2 + c

12. Mandelbrot Images

13. General 2-D Quadratic Map 100 % 10% 1% 0.1% Bounded solutions Chaotic solutions 0.1 1.0 10 a max

14. Probability of Chaotic Solutions Iterated maps Continuous flows (ODEs) 100% 10% 1% 0.1% 1 10 Dimension

15. Neural Net Architecture tanh

16. % Chaotic in Neural Networks

17. Types of Attractors Fixed Point Limit Cycle Torus Strange Attractor Spiral Radial

19. Stretching and Folding

20. Correlation Dimension 5 0.5 1 10 System Dimension Correlation Dimension

21. Lyapunov Exponent 1 10 System Dimension Lyapunov Exponent 10 1 0.1 0.01

22. Simplest Chaotic Flow d x /d t = y d y /d t = z d z /d t = - x + y 2 - Az 2.0168 < A < 2.0577

23. Simplest Chaotic Flow Attractor

24. Simplest Conservative Chaotic Flow x + x - x 2 = - 0.01 ... .

25. Chaotic Surrogate Models x n +1 = .671 - .416 x n - 1.014 x n 2 + 1.738 x n x n -1 +.836 x n -1 -.814 x n -1 2 Data Model Auto-correlation function (1/ f noise)

26. Aesthetic Evaluation